# Give applications of $\beth_3$ other than $\beth_3=\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N})))$

What are a few applications of $\beth_3$ other than $\beth_3=\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N})))$ or any other variations of generalized identities that can easily be found on Wikipedia? This is not a homework problem; I am merely studying independently.

To explain what I mean by “applications,” here is an application of each beth number before $\beth_3$ other than the power set definition:

• $\beth_0=\aleph_0$ is the basis of the beth numbers, and $\beth_0$ equals the cardinality of the natural numbers and also equals the cardinality of the integers. For concreteness, you could say that a series sums together this many terms.
• $\beth_1$ is the cardinality of the real numbers or any interval subset of the real line. You could loosely say that an integral adds together this many terms.
• While there are infinitely many functions that take a subset of the real line and return a subset of the real line (denoted $\mathbb{R^R}$), the number of them is defined an equals $\beth_2$.
• It's not true that $\beth_1$ is the cardinality of any subset of the real line - It's an upper bound for the cardinality of any subset of the real line. There are always finite/countable subsets of the real line and - should $\operatorname{CH}$ fail - there may be many more cardinalities which are realized by sets of real numbers. – Stefan Mesken Jul 30 '17 at 11:01
• @StefanMesken Ah, you are absolutely right. What I meant to communicate was that a continuous interval like $(0,1]$ or $[-4,2]$ has cardinality $\beth_1$. I hadn't stopped and realized that $\mathbb{N}\subset\mathbb{R}$. Thank you for catching that error! – Chase Ryan Taylor Jul 30 '17 at 16:58